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</head><body><div id="web_bg"></div><div id="sidebar"><div id="menu-mask"></div><div id="sidebar-menus"><div class="avatar-img is-center"><img src="/img/ywrby.png" onerror="onerror=null;src='/img/friend_404.gif'" alt="avatar"/></div><div class="site-data"><div class="data-item is-center"><div class="data-item-link"><a href="/archives/"><div class="headline">Articles</div><div class="length-num">127</div></a></div></div><div class="data-item is-center"><div class="data-item-link"><a href="/tags/"><div class="headline">Tags</div><div class="length-num">22</div></a></div></div></div><hr/><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页</span></a></div><div class="menus_item"><a class="site-page" href="/archives/"><i class="fa-fw fas fa-archive"></i><span> 归档</span></a></div><div class="menus_item"><a class="site-page" href="/tags/"><i class="fa-fw fas fa-tags"></i><span> 标签</span></a></div><div class="menus_item"><a class="site-page" href="javascript:void(0);"><i class="fa-fw fas fa-list"></i><span> Bilibili</span><i class="fas fa-chevron-down expand"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/bangumis/"><i class="fa-fw fas fa-video"></i><span> 追番列表</span></a></li><li><a class="site-page child" target="_blank" rel="noopener" href="https://space.bilibili.com/353923033"><i class="fa-fw fas fa-music"></i><span> B站主页</span></a></li></ul></div><div class="menus_item"><a class="site-page" href="/about/"><i class="fa-fw fas fa-heart"></i><span> 关于我</span></a></div></div></div></div><div class="post" id="body-wrap"><header class="post-bg" id="page-header" style="background-image: url('/img/72581222_p0.jpg')"><nav id="nav"><span id="blog_name"><a id="site-name" href="/">Ywrby个人博客网站</a></span><div id="menus"><div id="search-button"><a class="site-page social-icon search"><i class="fas fa-search fa-fw"></i><span> Search</span></a></div><div class="menus_items"><div class="menus_item"><a class="site-page" href="/"><i class="fa-fw fas fa-home"></i><span> 主页</span></a></div><div class="menus_item"><a class="site-page" href="/archives/"><i class="fa-fw fas fa-archive"></i><span> 归档</span></a></div><div class="menus_item"><a class="site-page" href="/tags/"><i class="fa-fw fas fa-tags"></i><span> 标签</span></a></div><div class="menus_item"><a class="site-page" href="javascript:void(0);"><i class="fa-fw fas fa-list"></i><span> Bilibili</span><i class="fas fa-chevron-down expand"></i></a><ul class="menus_item_child"><li><a class="site-page child" href="/bangumis/"><i class="fa-fw fas fa-video"></i><span> 追番列表</span></a></li><li><a class="site-page child" target="_blank" rel="noopener" href="https://space.bilibili.com/353923033"><i class="fa-fw fas fa-music"></i><span> B站主页</span></a></li></ul></div><div class="menus_item"><a class="site-page" href="/about/"><i class="fa-fw fas fa-heart"></i><span> 关于我</span></a></div></div><div id="toggle-menu"><a class="site-page"><i class="fas fa-bars fa-fw"></i></a></div></div></nav><div id="post-info"><h1 class="post-title">1-归并排序-算法复习</h1><div id="post-meta"><div class="meta-firstline"><span class="post-meta-date"><i class="far fa-calendar-alt fa-fw post-meta-icon"></i><span class="post-meta-label">Created</span><time class="post-meta-date-created" datetime="2021-11-07T10:40:55.000Z" title="Created 2021-11-07 18:40:55">2021-11-07</time><span class="post-meta-separator">|</span><i class="fas fa-history fa-fw post-meta-icon"></i><span class="post-meta-label">Updated</span><time class="post-meta-date-updated" datetime="2021-11-08T06:58:01.571Z" title="Updated 2021-11-08 14:58:01">2021-11-08</time></span></div><div class="meta-secondline"><span class="post-meta-separator">|</span><span class="post-meta-pv-cv" id="" data-flag-title="1-归并排序-算法复习"><i class="far fa-eye fa-fw post-meta-icon"></i><span class="post-meta-label">Post View:</span><span id="busuanzi_value_page_pv"></span></span></div></div></div></header><main class="layout" id="content-inner"><div id="post"><article class="post-content" id="article-container"><h1 id="归并"><a href="#归并" class="headerlink" title="归并"></a>归并</h1><p>要了解归并排序算法首先要了解归并这一过程，归并过程处理两个可比较数组（两个数组已经各自有序），在归并过程中，不断对两个数组的当前首元素进行比较，将较小的元素放置到新数组的下一位置。</p>
<h2 id="归并实现：-原地归并的抽象方法"><a href="#归并实现：-原地归并的抽象方法" class="headerlink" title="归并实现：(原地归并的抽象方法)"></a>归并实现：(原地归并的抽象方法)</h2><figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> cn.ywrby.test;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">MergeTest</span> </span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">boolean</span> <span class="title">less</span><span class="params">(Comparable v, Comparable w)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> v.compareTo(w) &lt; <span class="number">0</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="comment">//原地归并的抽象实现</span></span><br><span class="line">    <span class="comment">/*</span></span><br><span class="line"><span class="comment">    * 参数a表示已经部分有序的原数组（前一部分，后一部分分别有序）</span></span><br><span class="line"><span class="comment">    * 参数lo表示前一部分数组的首元素（前一部分最小值）</span></span><br><span class="line"><span class="comment">    * 参数mid表示前一部分数组最后一个元素（后一部分数组首元素的前一位，前一部分最大值）</span></span><br><span class="line"><span class="comment">    * 参数hi表示后一部分数组最后一位(后一部分最大值)</span></span><br><span class="line"><span class="comment">    * */</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">MergeTest</span><span class="params">(Comparable[] a,<span class="keyword">int</span> lo,<span class="keyword">int</span> mid,<span class="keyword">int</span> hi)</span></span>&#123;</span><br><span class="line">        <span class="keyword">int</span> i=lo,j=mid+<span class="number">1</span>;</span><br><span class="line">        Comparable[] aux=<span class="keyword">new</span> Comparable[hi+<span class="number">1</span>];</span><br><span class="line">        <span class="comment">//通过遍历复制原数组</span></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> k=lo;k&lt;=hi;k++)&#123;</span><br><span class="line">            aux[k]=a[k];</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//对原数组进行归并</span></span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> k=lo;k&lt;=hi;k++)&#123;</span><br><span class="line">            <span class="keyword">if</span>(i&gt;mid) a[k]=aux[j++];  <span class="comment">//若前一部分数组元素用尽，就取后一部分数组元素</span></span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span>(j&gt;hi) a[k]=aux[i++];  <span class="comment">//若后一部分数组元素，就取前一部分数组元素</span></span><br><span class="line">            <span class="comment">// 两个都没有用尽，就比较两数组当前首元素大小，取二者中较小的</span></span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span>(less(aux[j],aux[i])) a[k]=aux[j++];  </span><br><span class="line">            <span class="keyword">else</span> a[k]=aux[i++];  </span><br><span class="line">        &#125;</span><br><span class="line">    &#125; </span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>

<hr>
<h1 id="自顶向下的归并排序"><a href="#自顶向下的归并排序" class="headerlink" title="自顶向下的归并排序"></a>自顶向下的归并排序</h1><p>基于原地归并的抽象实现完成的一种递归排序，首先不断对原数组进行分割，直至不能分割（每个数组中仅含一个元素），然后以每两个数组进行归并（因为只有一个元素，所以数组有序），经过一轮归并，数组中元素为2个或1个，继续递归进行归并排序，直至数组全部归并只剩一个</p>
<p>整个过程利用了分治思想，将一个大问题拆解为若干个简单的小问题加以解决</p>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br><span class="line">85</span><br><span class="line">86</span><br><span class="line">87</span><br><span class="line">88</span><br><span class="line">89</span><br><span class="line">90</span><br><span class="line">91</span><br><span class="line">92</span><br><span class="line">93</span><br><span class="line">94</span><br><span class="line">95</span><br><span class="line">96</span><br><span class="line">97</span><br><span class="line">98</span><br><span class="line">99</span><br><span class="line">100</span><br><span class="line">101</span><br><span class="line">102</span><br><span class="line">103</span><br><span class="line">104</span><br><span class="line">105</span><br><span class="line">106</span><br><span class="line">107</span><br><span class="line">108</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> cn.ywrby.sorts;</span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> cn.ywrby.tools.StopWatch;</span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> java.util.Random;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment">//自顶向下的归并排序</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">MergeSort</span> </span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">boolean</span> <span class="title">less</span><span class="params">(Comparable v, Comparable w)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> v.compareTo(w) &lt; <span class="number">0</span>;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="keyword">private</span> <span class="keyword">static</span> Comparable[] aux;</span><br><span class="line"></span><br><span class="line">    <span class="comment">//原地归并的抽象实现</span></span><br><span class="line">    <span class="comment">/*</span></span><br><span class="line"><span class="comment">     * 参数a表示已经部分有序的原数组（前一部分，后一部分分别有序）</span></span><br><span class="line"><span class="comment">     * 参数lo表示前一部分数组的首元素（前一部分最小值）</span></span><br><span class="line"><span class="comment">     * 参数mid表示前一部分数组最后一个元素（后一部分数组首元素的前一位，前一部分最大值）</span></span><br><span class="line"><span class="comment">     * 参数hi表示后一部分数组最后一位(后一部分最大值)</span></span><br><span class="line"><span class="comment">     * */</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">merge</span><span class="params">(Comparable[] a, <span class="keyword">int</span> lo, <span class="keyword">int</span> mid, <span class="keyword">int</span> hi)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> i = lo, j = mid + <span class="number">1</span>;</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> k = lo; k &lt;= hi; k++) &#123;</span><br><span class="line">            aux[k] = a[k];</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//对原数组进行归并</span></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> k = lo; k &lt;= hi; k++) &#123;</span><br><span class="line">            <span class="keyword">if</span> (i &gt; mid) a[k] = aux[j++];  <span class="comment">//若前一部分数组元素用尽，就取后一部分数组元素</span></span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span> (j &gt; hi) a[k] = aux[i++];  <span class="comment">//若后一部分数组元素，就取前一部分数组元素</span></span><br><span class="line">                <span class="comment">// 两个都没有用尽，就比较两数组当前首元素大小，取二者中较小的</span></span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span> (less(aux[j], aux[i])) a[k] = aux[j++];</span><br><span class="line">            <span class="keyword">else</span> a[k] = aux[i++];</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="comment">//准备额外空间用于盛放排序后的数组</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">sort</span><span class="params">(Comparable[] a)</span> </span>&#123;</span><br><span class="line">        aux = <span class="keyword">new</span> Comparable[a.length];</span><br><span class="line">        sort(a, <span class="number">0</span>, a.length - <span class="number">1</span>);</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="comment">/*基于原地归并的抽象实现完成的一种递归排序</span></span><br><span class="line"><span class="comment">     *首先不断对原数组进行分割，直至不能分割（每个数组中仅含一个元素）</span></span><br><span class="line"><span class="comment">     *然后以每两个数组进行归并（因为只有一个元素，所以数组有序）</span></span><br><span class="line"><span class="comment">     *经过一轮归并，数组中元素为2个或1个</span></span><br><span class="line"><span class="comment">     *继续递归进行归并排序，直至数组全部归并只剩一个</span></span><br><span class="line"><span class="comment">     */</span></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">sort</span><span class="params">(Comparable[] a, <span class="keyword">int</span> lo, <span class="keyword">int</span> hi)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">if</span> (hi &lt;= lo) <span class="keyword">return</span>;</span><br><span class="line">        <span class="keyword">int</span> mid = lo + (hi - lo) / <span class="number">2</span>;</span><br><span class="line">        sort(a, lo, mid);</span><br><span class="line">        sort(a, mid + <span class="number">1</span>, hi);</span><br><span class="line">        merge(a, lo, mid, hi);</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">exch</span><span class="params">(Comparable[] a, <span class="keyword">int</span> i, <span class="keyword">int</span> j)</span> </span>&#123;</span><br><span class="line">        Comparable temp = a[i];</span><br><span class="line">        a[i] = a[j];</span><br><span class="line">        a[j] = temp;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">show</span><span class="params">(Comparable[] a)</span> </span>&#123;</span><br><span class="line">        System.out.print(<span class="string">&quot;After sort : &quot;</span>);</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; a.length; i++) &#123;</span><br><span class="line">            System.out.print(a[i] + <span class="string">&quot;  &quot;</span>);</span><br><span class="line">        &#125;</span><br><span class="line">        System.out.println();</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">boolean</span> <span class="title">isSorted</span><span class="params">(Comparable[] a)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; a.length - <span class="number">1</span>; i++) &#123;</span><br><span class="line">            <span class="keyword">if</span> (!less(a[i], a[i + <span class="number">1</span>])) &#123;</span><br><span class="line">                <span class="keyword">return</span> <span class="keyword">false</span>;</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> <span class="keyword">true</span>;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> N = <span class="number">1000</span>;</span><br><span class="line">        Comparable&lt;Double&gt;[] test = <span class="keyword">new</span> Comparable[N];</span><br><span class="line">        System.out.print(<span class="string">&quot;before sort : &quot;</span>);</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; N; i++) &#123;</span><br><span class="line">            <span class="keyword">double</span> data = Math.random();</span><br><span class="line">            System.out.print(data + <span class="string">&quot;  &quot;</span>);</span><br><span class="line">            test[i] = data;</span><br><span class="line">        &#125;</span><br><span class="line">        System.out.println();</span><br><span class="line">        StopWatch watch = <span class="keyword">new</span> StopWatch();</span><br><span class="line">        sort(test);</span><br><span class="line">        <span class="keyword">double</span> time = watch.elapsedTime();</span><br><span class="line">        <span class="comment">/*</span></span><br><span class="line"><span class="comment">         * assert关键字：assert [boolean 表达式]</span></span><br><span class="line"><span class="comment">         * 如果[boolean表达式]为true，则程序继续执行。</span></span><br><span class="line"><span class="comment">         * 如果为false，则程序抛出AssertionError，并终止执行。</span></span><br><span class="line"><span class="comment">         */</span></span><br><span class="line">        <span class="function"><span class="keyword">assert</span> <span class="title">isSorted</span><span class="params">(test)</span></span>;</span><br><span class="line">        show(test);</span><br><span class="line">        System.out.println(<span class="string">&quot;time=&quot;</span> + time);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br></pre></td></tr></table></figure>

<h2 id="算法分析"><a href="#算法分析" class="headerlink" title="算法分析"></a>算法分析</h2><p>对于长度为N的任意数组，自顶向下的归并排序需要NlgN/2~NlgN次++比较++</p>
<p>令C(N)表示一个长度为N的数组需要进行比较的次数。易知：C(0)=C(1)=0,且自顶向下排序采用了递归的方法，所以可以写成：<br>$$C(N)&lt;=C_{前}(N/2)+C_{后}(N/2)+N$$</p>
<p>第一项表示数组前半部分比较次数，第二项则表示后半部分比较次数，最后一项表示将两项归并到一起所需要的最大比较次数</p>
<p>$$C(N)&gt;=C_{前}(N/2)+C_{后}(N/2)+N/2$$<br>同理最后一项表示归并时最小比较次数</p>
<p>以$N=2^n$时为例下的++最坏情况++进行分析，可以得到如下结论：</p>
<p>$$C(2^n)=C_{前}(2^{n-1})+C_{后}(2^{n-1})+2^n=2*C(2^{n-1})+2^n$$<br>将上式两边同时除以2^n得到：</p>
<p>$$C(2^n)/2^n=C(2^{n-1})/2^{n-1}+1$$<br>利用该式可以替换右边第一项得到：</p>
<p>$$C(2^n)/2^n=C(2^{n-2})/2^{n-2}+1+1$$<br>重复n次得到<br>$$C(2^n)/2^n=C(0)/2^{0}+n=n$$<br>因此</p>
<p>$$C(2^n)=n<em>2^n$$<br>进而由N=2^n得到<br>$$C(N)=N</em>lgN$$</p>
<p>虽然这是对特殊情况的一种讨论，但我们不难理解这对任意N是普遍适用的</p>
<hr>
<p>对于长度为N的任意数组，自顶向下的归并排序最多需要++访问数组++6NlgN次</p>
<p>每次归并最多访问数组6N次（第一个for循环的复制过程2N次，比较过程中最多2N次，将排序好的元素放回2N次），所以由上一个命题易知，最多需要访问数组6NlgN次</p>
<hr>
<h3 id="优点"><a href="#优点" class="headerlink" title="优点"></a>优点</h3><p>运行时间与NlgN成正比，所以可以处理数百万甚至更大规模数组，这是初级排序算法无法做到的</p>
<h3 id="缺点"><a href="#缺点" class="headerlink" title="缺点"></a>缺点</h3><p>辅助数组所使用的额外空间与N成正比</p>
<hr>
<h3 id="算法优化"><a href="#算法优化" class="headerlink" title="算法优化"></a>算法优化</h3><ol>
<li>对小规模数组使用插入排序而不是始终递归</li>
<li>添加方法以测试数组是否已经有序(a[mid]&lt;=a[mid+1])</li>
<li>不将元素复制到辅助数组</li>
</ol>
<hr>
<h1 id="自底向上的归并排序"><a href="#自底向上的归并排序" class="headerlink" title="自底向上的归并排序"></a>自底向上的归并排序</h1><p>自底向上的归并排序会遍历整个数组，根据子数组大小进行两两排序。子数组的大小sz的初始值为1，每次加倍。最后一个字数组的大小只有在数组大小是sz的偶数倍的时候才会等于sz(否则会比sz小)</p>
<figure class="highlight java"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br><span class="line">46</span><br><span class="line">47</span><br><span class="line">48</span><br><span class="line">49</span><br><span class="line">50</span><br><span class="line">51</span><br><span class="line">52</span><br><span class="line">53</span><br><span class="line">54</span><br><span class="line">55</span><br><span class="line">56</span><br><span class="line">57</span><br><span class="line">58</span><br><span class="line">59</span><br><span class="line">60</span><br><span class="line">61</span><br><span class="line">62</span><br><span class="line">63</span><br><span class="line">64</span><br><span class="line">65</span><br><span class="line">66</span><br><span class="line">67</span><br><span class="line">68</span><br><span class="line">69</span><br><span class="line">70</span><br><span class="line">71</span><br><span class="line">72</span><br><span class="line">73</span><br><span class="line">74</span><br><span class="line">75</span><br><span class="line">76</span><br><span class="line">77</span><br><span class="line">78</span><br><span class="line">79</span><br><span class="line">80</span><br><span class="line">81</span><br><span class="line">82</span><br><span class="line">83</span><br><span class="line">84</span><br></pre></td><td class="code"><pre><span class="line"><span class="keyword">package</span> cn.ywrby.sorts;</span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> cn.ywrby.tools.StopWatch;</span><br><span class="line"></span><br><span class="line"><span class="comment">//自底向上的归并排序</span></span><br><span class="line"></span><br><span class="line"><span class="keyword">import</span> java.util.Random;</span><br><span class="line"></span><br><span class="line"><span class="keyword">public</span> <span class="class"><span class="keyword">class</span> <span class="title">MergeBuSort</span> </span>&#123;</span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">boolean</span> <span class="title">less</span><span class="params">(Comparable v, Comparable w)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">return</span> v.compareTo(w) &lt; <span class="number">0</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    </span><br><span class="line">    <span class="keyword">private</span> <span class="keyword">static</span> Comparable[] aux;</span><br><span class="line">    </span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">merge</span><span class="params">(Comparable[] a, <span class="keyword">int</span> lo, <span class="keyword">int</span> mid, <span class="keyword">int</span> hi)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> i = lo, j = mid + <span class="number">1</span>;</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> k = lo; k &lt;= hi; k++) &#123;</span><br><span class="line">            aux[k] = a[k];</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="comment">//对原数组进行归并</span></span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> k = lo; k &lt;= hi; k++) &#123;</span><br><span class="line">            <span class="keyword">if</span> (i &gt; mid) a[k] = aux[j++];  <span class="comment">//若前一部分数组元素用尽，就取后一部分数组元素</span></span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span> (j &gt; hi) a[k] = aux[i++];  <span class="comment">//若后一部分数组元素，就取前一部分数组元素</span></span><br><span class="line">                <span class="comment">// 两个都没有用尽，就比较两数组当前首元素大小，取二者中较小的</span></span><br><span class="line">            <span class="keyword">else</span> <span class="keyword">if</span> (less(aux[j], aux[i])) a[k] = aux[j++];</span><br><span class="line">            <span class="keyword">else</span> a[k] = aux[i++];</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">sort</span><span class="params">(Comparable[] a)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">int</span> N=a.length;</span><br><span class="line">        aux=<span class="keyword">new</span> Comparable[a.length];</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> sz=<span class="number">1</span>;sz&lt;N;sz*=<span class="number">2</span>)&#123;  <span class="comment">//sz：子数组大小</span></span><br><span class="line">            <span class="keyword">for</span>(<span class="keyword">int</span> lo=<span class="number">0</span>;lo&lt;N-sz;lo+=sz*<span class="number">2</span>)&#123;  <span class="comment">//lo：子数组索引</span></span><br><span class="line">                merge(a,lo,lo+sz-<span class="number">1</span>,Math.min(lo+sz*<span class="number">2</span>-<span class="number">1</span>,N-<span class="number">1</span>));</span><br><span class="line">            &#125;</span><br><span class="line">        &#125;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    </span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">exch</span><span class="params">(Comparable[] a, <span class="keyword">int</span> i, <span class="keyword">int</span> j)</span> </span>&#123;</span><br><span class="line">        Comparable temp = a[i];</span><br><span class="line">        a[i] = a[j];</span><br><span class="line">        a[j] = temp;</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">private</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">show</span><span class="params">(Comparable[] a)</span> </span>&#123;</span><br><span class="line">        System.out.print(<span class="string">&quot;After sort : &quot;</span>);</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; a.length; i++) &#123;</span><br><span class="line">            System.out.print(a[i] + <span class="string">&quot;  &quot;</span>);</span><br><span class="line">        &#125;</span><br><span class="line">        System.out.println();</span><br><span class="line">    &#125;</span><br><span class="line"></span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">boolean</span> <span class="title">isSorted</span><span class="params">(Comparable[] a)</span> </span>&#123;</span><br><span class="line">        <span class="keyword">for</span> (<span class="keyword">int</span> i = <span class="number">0</span>; i &lt; a.length-<span class="number">1</span>; i++) &#123;</span><br><span class="line">            <span class="keyword">if</span> (!less(a[i],a[i+<span class="number">1</span>]))&#123;<span class="keyword">return</span> <span class="keyword">false</span>;&#125;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">return</span> <span class="keyword">true</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="function"><span class="keyword">public</span> <span class="keyword">static</span> <span class="keyword">void</span> <span class="title">main</span><span class="params">(String[] args)</span></span>&#123;</span><br><span class="line">        <span class="keyword">int</span> N=<span class="number">100</span>;</span><br><span class="line">        Comparable&lt;Double&gt;[] test=<span class="keyword">new</span> Comparable[N];</span><br><span class="line">        System.out.print(<span class="string">&quot;before sort : &quot;</span>);</span><br><span class="line">        <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;N;i++)&#123;</span><br><span class="line">            <span class="keyword">double</span> data=Math.random();</span><br><span class="line">            System.out.print(data+<span class="string">&quot;  &quot;</span>);</span><br><span class="line">            test[i]=data;</span><br><span class="line">        &#125;</span><br><span class="line">        System.out.println();</span><br><span class="line">        StopWatch watch=<span class="keyword">new</span> StopWatch();</span><br><span class="line">        sort(test);</span><br><span class="line">        <span class="keyword">double</span> time=watch.elapsedTime();</span><br><span class="line">        <span class="comment">/*</span></span><br><span class="line"><span class="comment">         * assert关键字：assert [boolean 表达式]</span></span><br><span class="line"><span class="comment">         * 如果[boolean表达式]为true，则程序继续执行。</span></span><br><span class="line"><span class="comment">         * 如果为false，则程序抛出AssertionError，并终止执行。</span></span><br><span class="line"><span class="comment">         */</span></span><br><span class="line">        <span class="function"><span class="keyword">assert</span> <span class="title">isSorted</span><span class="params">(test)</span></span>;</span><br><span class="line">        show(test);</span><br><span class="line">        System.out.println(<span class="string">&quot;time=&quot;</span>+time);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>


<h1 id="归并排序的局限性"><a href="#归并排序的局限性" class="headerlink" title="归并排序的局限性"></a>归并排序的局限性</h1><ol>
<li>归并排序的空间复杂度不是最优的</li>
<li>除了比较，算法的其他操作（访问数组）也可能很重要</li>
<li>不进行比较也能将某些数据排序</li>
</ol>
</article><div class="post-copyright"><div class="post-copyright__author"><span class="post-copyright-meta">Author: </span><span class="post-copyright-info"><a href="mailto:undefined">Ywrby</a></span></div><div class="post-copyright__type"><span class="post-copyright-meta">Link: </span><span class="post-copyright-info"><a href="http://ywrby.cn/2021/11/07/1-%E5%BD%92%E5%B9%B6%E6%8E%92%E5%BA%8F-%E7%AE%97%E6%B3%95%E5%A4%8D%E4%B9%A0/">http://ywrby.cn/2021/11/07/1-%E5%BD%92%E5%B9%B6%E6%8E%92%E5%BA%8F-%E7%AE%97%E6%B3%95%E5%A4%8D%E4%B9%A0/</a></span></div><div class="post-copyright__notice"><span class="post-copyright-meta">Copyright Notice: </span><span class="post-copyright-info">All articles in this blog are licensed under <a target="_blank" rel="noopener" href="https://creativecommons.org/licenses/by-nc-sa/4.0/">CC BY-NC-SA 4.0</a> unless stating additionally.</span></div></div><div class="tag_share"><div class="post-meta__tag-list"><a class="post-meta__tags" 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href="https://www.facebook.com/profile.php?id=100033741068822" target="_blank" title="Facebook"><i class="fab fa-facebook"></i></a></div></div><div class="card-widget card-announcement"><div class="item-headline"><i class="fas fa-bullhorn card-announcement-animation"></i><span>Announcement</span></div><div class="announcement_content">Less is more</div></div><div class="sticky_layout"><div class="card-widget" id="card-toc"><div class="item-headline"><i class="fas fa-stream"></i><span>Catalog</span></div><div class="toc-content"><ol class="toc"><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%BD%92%E5%B9%B6"><span class="toc-number">1.</span> <span class="toc-text">归并</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E5%BD%92%E5%B9%B6%E5%AE%9E%E7%8E%B0%EF%BC%9A-%E5%8E%9F%E5%9C%B0%E5%BD%92%E5%B9%B6%E7%9A%84%E6%8A%BD%E8%B1%A1%E6%96%B9%E6%B3%95"><span class="toc-number">1.1.</span> <span class="toc-text">归并实现：(原地归并的抽象方法)</span></a></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E8%87%AA%E9%A1%B6%E5%90%91%E4%B8%8B%E7%9A%84%E5%BD%92%E5%B9%B6%E6%8E%92%E5%BA%8F"><span class="toc-number">2.</span> <span class="toc-text">自顶向下的归并排序</span></a><ol class="toc-child"><li class="toc-item toc-level-2"><a class="toc-link" href="#%E7%AE%97%E6%B3%95%E5%88%86%E6%9E%90"><span class="toc-number">2.1.</span> <span class="toc-text">算法分析</span></a><ol class="toc-child"><li class="toc-item toc-level-3"><a class="toc-link" href="#%E4%BC%98%E7%82%B9"><span class="toc-number">2.1.1.</span> <span class="toc-text">优点</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%BC%BA%E7%82%B9"><span class="toc-number">2.1.2.</span> <span class="toc-text">缺点</span></a></li><li class="toc-item toc-level-3"><a class="toc-link" href="#%E7%AE%97%E6%B3%95%E4%BC%98%E5%8C%96"><span class="toc-number">2.1.3.</span> <span class="toc-text">算法优化</span></a></li></ol></li></ol></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E8%87%AA%E5%BA%95%E5%90%91%E4%B8%8A%E7%9A%84%E5%BD%92%E5%B9%B6%E6%8E%92%E5%BA%8F"><span class="toc-number">3.</span> <span class="toc-text">自底向上的归并排序</span></a></li><li class="toc-item toc-level-1"><a class="toc-link" href="#%E5%BD%92%E5%B9%B6%E6%8E%92%E5%BA%8F%E7%9A%84%E5%B1%80%E9%99%90%E6%80%A7"><span class="toc-number">4.</span> <span class="toc-text">归并排序的局限性</span></a></li></ol></div></div><div class="card-widget card-recent-post"><div class="item-headline"><i class="fas fa-history"></i><span>Recent Post</span></div><div class="aside-list"><div class="aside-list-item"><a class="thumbnail" href="/2021/12/25/22-%E5%86%85%E5%AD%98%E7%A9%BA%E9%97%B4%E6%89%A9%E5%85%85%EF%BC%88%E8%A6%86%E7%9B%96%E4%B8%8E%E4%BA%A4%E6%8D%A2%EF%BC%89/" title="22-内存空间扩充（覆盖与交换）"><img src="/img/67888138_p0.jpg" onerror="this.onerror=null;this.src='/img/404.jpg'" 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